Digital Communications

Digital Communications Chapter 1. Introduction Po-Ning Chen, Professor Institute of Communications Engineering National Chiao-Tung University, Taiwan Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 1/ 22

Digital Communications What we study in this course is: Theories of information transmission in digital form from oneormoresourcestooneormoredestinations. Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 2/ 22

1.1 Elements of digital communication system Functional diagram of a digital communication system Information source and input transducer Source encoder Channel encoder Digital modulator Channel Output signal Output transducer Source decoder Channel decoder Digital demodulator Basic elements of a digital communication system Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 3/ 22

1.2 Comm channels and their characteristics Physical channel media (magnetic-electrical signaled) Wireline channel Telephone line, twisted-pair and coaxial cable, etc. (modulated light beam) Fiber-optical channel (antenna radiated) Wireless electromagnetic channel ground-wave propagation, sky-wave propagation, line-of-sight (LOS) propagation, etc. (multipath) Underwater acoustic channel... etc. Virtual channel Storage channel Magnetic storage, CD, DVD, etc. Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 4/ 22

1.2 Comm channels and their characteristics Channel impairments Thermal noise (additive noise) Signal attenuation Amplitude and phase distortion Multi-path distortion Limitations of channel usage Transmission power Receiver sensitivity Bandwidth Transmission time Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 5/ 22

1.3 Math models for communication channels Additive noise channel (with attenuation) In studying these channels, a mathematical model is necessary. Channel r(t)=αs(t)+n(t) s(t) r(t) α n(t) where α is the attenuation factor s(t) is the transmitted signal n(t) is the additive random noise (a random process, usually Gaussian) Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 6/ 22

1.3 Math models for communication channels Linear filter channel with additive noise To meet the specified bandwidth limitation s(t) Linear time-invariant filter c(t) Channel n(t) r(t) r(t) = s(t) c(t)+n(t) = c(τ)s(t τ) dτ + n(t) Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 7/ 22

1.3 Math models for communication channels Linear time-variant (LTV) filter channel with additive noise s(t) Linear time-variant filter c(τ; t) Channel r(t) n(t) r(t) = s(t) c(τ; t)+n(t) = c(τ; t)s(t τ) dτ + n(t) τ is the argument for filtering. t is the argument for time-dependence. The time-invariant filter can be viewed as a special case of the time-variant filter. Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 8/ 22

Assume n(t)=0 (noise-free). LTI versus LTV Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 9/ 22

1.3 Math models for communication channels LTV filter channel with additive noise c(τ; t) usually has the form c(τ; t) = L k=1 a k (t)δ(τ τ k ) where {a k (t)} L k=1 represent the possibly time-varying attenuation factor for the L multipath propagation paths {τ k } L k=1 are the corresponding time delays. Hence L r(t) = a k (t)s(t τ k )+n(t) k=1 Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 10/ 22

1.3 Math models for communication channels Time varying multipath fading channel r(t) = a 1 (t)s(t τ 1 )+a 2 (t)s(t τ 2 )+a 3 (t)s(t τ 3 )+n(t) Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 11/ 22

1.4 A historical perspective in the developement of digital communications Morse code (1837) Variable-length binary code for telegraph Baudot code (1875) Fixed-length binary code of length 5 Nyquist (1924) Determine the maximum signaling rate without intersymbol interference over, e.g., a telegraph channel Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 12/ 22

1.4 A historical perspective in the developement of digital communications - Nyquist rate Nyquist (1924) Define basic pulse shape g(t) that is bandlimited to W. One wishes to transmit { 1, 1} signals in terms of g(t), or equivalently, one wishes to transmit a 0, a 1, a 2,... in { 1, 1} in terms of s(t) defined as s(t)=a 0 g(t)+a 1 g(t T )+a 2 g(t 2T )+ Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 13/ 22

1.4 A historical perspective in the developement of digital communications - Nyquist rate Example. (a 0, a 1, a 2,...)=(+1, 1, +1,...). Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 14/ 22

1.4 A historical perspective in the developement of digital communications - Nyquist rate Question that Nyquist shoots for: What is the maximum rate that the data can be transmitted under the constraint that g(t) causes no intersymbol interference (at the sampling instances)? Answer : 2W pulses/second. (Not 2W bits/second!) What g(t) can achieve this rate? Answer : g(t)= sin(2πwt). 2πWt Conclusion: A bandlimited-to-w basic pulse shape signal (or symbol) can convey at most 2W pulses/second (or symbols/second) without introducing inter-pulse (or inter-symbol) interference. Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 15/ 22

1.4 A historical perspective in the developement of digital communications - Sampling theorem Shannon (1948) Sampling theorem A signal of bandwidth W can be reconstructed from samples taken at the Nyquist rate (= 2W samples/second) using the interpolation formula s(t) = n= s ( n 2W (t n/(2w ))] ) (sin[2πw ). 2πW (t n/(2w )) Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 16/ 22

1.4 A historical perspective in the developement of digital communications - Shannon s channel coding theorem Channel capacity of additive white Gaussian noise where C = W log 2 (1 + P WN 0 ) bits/second W is the bandwidth of the bandlimited channel, P is the average transmitted power, N 0 is single-sided noise power per hertz. Shannon s channel coding theorem Let R be the information rate of the source. Then if R < C, it is theoretically possible to achieve reliable (asymptotically error-free) transmission by appropriate coding; if R > C, reliable transmission is impossible. This gives birth to a new field named Information Theory. Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 17/ 22

1.4 A historical perspective in the developement of digital communications Other important contributions Harvey (1928), based on Nyquists result, concluded that A maximum reliably transmitted data rate exists for a bandlimited channel under maximum transmitted signal amplitude constraint and minimum transmitted signal amplitude resolution constraint. Kolmogorov (1939) and Wiener (1942) Optimum linear (Kolmogorov-Wiener) filter whose output is the best mean-square approximation to the desired signal s(t) in presence of additive noise. Kotenikov (1947), Wozencraft and Jacobs (1965) Use geometric approach to analyze various coherent digital communication systems. Hamming (1950) Hamming codes Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 18/ 22

1.4 A historical perspective in the developement of digital communications Other important contributions (Continue) Muller (1954), Reed (1954), Reed and Solomon (1960), Bose and Ray-Chaudhuri (1960), and Goppa (1970,1971) New block codes, such as Reed-Solomon codes, Bose-Chaudhuri-Hocquenghem (BCH) codes and Goppa codes. Forney (1966) Concatenated codes Chien (1964), Berlekamp (1968) Berlekamp-Massey BCH-code decoding algorithm Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 19/ 22

1.4 A historical perspective in the developement of digital communications Other important contributions (Continue) Wozencraft and Reiffen (1961), Fano (1963), Zigangirov (1966), Jelinek (1969), Forney (1970, 1972, 1974) and Viterbi (1967, 1971) Convolusional code and its decoding Ungerboeck (1982), Forney et al. (1984), Wei (1987) Trellis-coded modulation Ziv and Lempel (1977, 1978) and Linde et al. (1980) Source encoding and decoding algorithms, such as Lempel-Ziv code Berrou et al. (1993) Turbo code and iterative decoding Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 20/ 22

1.4 A historical perspective in the developement of digital communications Other important contributions (Continue) Gallager (1963), Davey and Mackay (1998) Low-density parity-check code and the sum-product decoding algorithm Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 21/ 22

What you learn from Chapter 1 Mathematical models of time-variant and time-invariant additive noise channels multipath channels Nyquist rates and Sampling theorem Digital Communications: Chapter 1 Ver. 2015.10.19 Po-Ning Chen 22/ 22